The literary works of Lewis Carroll are littered with references to mathematics, the vast majority more subtle and involved than the above. Although unquestionably written for a young audience, his renowned Alice books and his less well known novels all include references and allusions to sophisticated mathematical concepts and problems. This essay will introduce Carroll as a man and author, and attempt to identify examples of these references, analyzing their purpose and effect.

Charles Lutwidge Dodgson was born on January 271832 in rural Cheshire, the third of eleven children. He was apparently of colourful ancestry, Bakewell (1996) claiming he can, with “a certain amount of ingenuity” claim to have been related to Lady Godiva, the Earls of Mercia and Northumbria and even Queen Victoria. Yet the man himself seems to have led a far from colourful life. In 1851 he entered Christ Church, Oxford where he distinguished himself in mathematics, and remained there as librarian, mathematics lecturer, and curator for the next forty-seven years.

As a mathematician his texts are widely considered as conservative and never significant, and are read now only because of historical interest. As a man he has been traditionally regarded as eccentric and at home only in the company of little girls. In fact, only the recent work of Karoline Leach (1996) in destroying the reliability of Dodgson’s early biographers has begun to alter the long held view that his love for young girls was a suppressed sexual passion.

Under the pen name of Lewis Carroll, Dodgson produced in Alice’s Adventures in Wonderland, and the follow up Through the Looking Glass and what Alice Found There, two of the best loved children’s books of all. It is widely known that the Alice tales were born during what Dodgson escribed in his diary as “that golden afternoon” of the 4th of July 1862 when he entertained the children of the Dean at Christ Church with his stories, whilst on a rowing trip in Oxford. One of these children was of course the six-year-old Alice Liddell, the inspiration for the main character.

Some months later Dodgson was encouraged to write the first draft of Alice’s Adventures Under Ground, which by 1865 had developed into Alice’s Adventures in Wonderland. This being well received, he followed up with the equally successful Through the Looking Glass in 1871. His first ublication of humorous and other verses was Phantasmagoria and Other Poems in 1869. The acclaimed nonsense poem The Hunting of the Snark: an Agony in Eight Fits was published in March 1876.

Sylvie and Bruno in 1889 and Sylvie and Bruno Concluded four years later represented the biggest effort by Carroll in literature, a mixture of fairy-tale, social novel and collection of ethical discussions, but they never came close to matching the success of his Alice books. His other works are generally not well known, although were often quite interesting. A Tangled Tale published in 1885 was a series of puzzles and aradoxes and according to Fisher (1972) represented “… Carroll at his playful best in combining tantalising whimsy with straightforward mathematics”.

Similar puzzles appeared in Pillow-Problems in 1893. An attempt to introduce formal deductive logic to a kindergarten audience, was The Game of Logic which Carroll published in 1886. Although unsuccessful in so far as it was too challenging for such a young audience, it did introduce some interesting methods and the examples were as quaint as the contents of Wonderland. In developing the theme for a more adult audience his last book was Symbolic Logic: Part 1, Elementary, interestingly under he authorship of Lewis Carroll rather his real name that he still used for his mathematical publications.

Of these none were of great impact, Euclid and his modern rivals is usually pointed to as a work of historical interest if not great mathematical incision. To consider first Alice’s Adventure in Wonderland, mathematical references and hints are present right from the very beginning. Alice’s fall down the rabbit hole at the start of Wonderland is prompted partly by her dissatisfaction her sister’s book which “… had no pictures or conversations in it, ‘and what is the use of a book,’ thought Alice, without pictures or conversation? ‘”.

From a mathematical point of view it is interesting that from within Alice’s system of understanding (that of a seven-year-old girl), her sister’s book has no use in that it contains nothing by which she is able to interpret any meaning, through mechanically applying her rules of inference. Alice could just as easily have followed the white rabbit directly into the “low long hall” without falling four thousand miles, but it is a device by which the continued theme of distortion of time is introduced; without it the Mad Hatter’s watch could not possibly be two days out.

Not only that, ut it allows for the introduction of a bizarre mechanics problem, seemingly a passion of Carroll’s. Speculation of the outcome of a fall through a ‘hole’ in the centre of the earth, assuming one to be possible, continued to be rife in Carroll’s day having been previously addressed by Plutarch, Bacon and Voltaire. That Carroll used such a method to introduce Alice to her Wonderland is typical of the way in which he hints at his mathematical enthusiasms throughout his works..

We can see though his other publications that he was interested in such dynamics problems, that through his literature he was willing to ponder the effects of gravity in imilarly unfeasible, yet mathematically interesting conditions. There is for example Mein Herr’s gravity-trains in Sylvie and Bruno Concluded, which are “without any engines-nothing is needed but machinery to stop them with”.

“‘Can you explain the process? ‘ said Lady Muriel. ‘Without using that language, which I can’t speak fluently? ‘ ‘Easily,’ said Mein Herr. Each railway is in a long tunnel, perfectly straight: so of course the middle of it is nearer the centre of the globe than the two ends: so every train runs half-way down-hill, and that gives it force enough to run the other half up-hill. ‘” Here we can clearly see the mathematical brain of Carroll, or rather Professor Dodgson, at work. Perhaps such a system might under specific conditions in theory behave as Mein Herr describes, yet the idea finds itself in such a ‘little fairy-tale’ (as Carroll describes the book in its preface) because as everybody knows in a real world of friction and other external forces the idea is fantasy.

Yet, this fantasy situation has been arrived at through following a reasonable concept to an unreasonable length. Meanwhile Alice would presumably neglecting forces due to the earth’s rotation, perform simple harmonic motion about the centre of the arth damped by air resistance until eventually coming to rest at the centre of the earth. Yet here Carroll is not interested in a discussion of the cleverness of the ideas of such dynamics, but for us to share in Alice’s bewilderment, and so she safely lands “… upon a heap of dry leaves, and the fall was over”.

A recurring theme in Carroll’s works, is that of geographic maps. As Alice falls down the rabbit hole she points out “maps and pictures hung upon pegs”. Just as a symbol in algebra is an arbitrary encoding of something, a map is similarly an arbitrary encoding of a portion of geographical errain. Carroll turns his attention to maps in a particularly interesting section from Sylvie and Bruno, when the ever knowledgeable Mein Herr explains the development of map-making in his country. “`We very soon got to six yards to the mile.

Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile! ‘ `Have you used it much? ‘ I enquired. `It has never been spread out, yet,’ said Mein Herr: `the farmers objected: they said it would cover the whole country, and shut out the sunlight! So e now use the country itself, as its own map, and I assure you it does nearly as well. ” The idea of a map of unit size, such that it would cover exactly the country it is supposed to represent, appears ludicrous to us.

Is it obviously ridiculously impractical, and of no real use or interest. However in mathematical terms, Carroll has taken the one-to-one correspondence of the map and terrain it describes, and made it an identity function. This concept is perfectly valid and interesting in the world of mathematics, only when giving it this physical meaning does it appear at all absurd. In constructing his make-believe world Carroll seems to be strictly applying the notions of abstract mathematics to elements of reality, as a means of creating the dreamlike state his characters inhabit.

A similarly strange and useless map appears in The Hunting of the Snark: an Agony in Eight Fits. “He had bought a large map representing the sea, Without the least vestige of land: And the crew were much pleased when they found it to be A map they could all understand. ” The Bellman has, it is revealed, bought “A perfect and absolute blank! “. Despite the obvious difficulties a blank map will bring, the buying what as defined as a map has given the Bellman and his crew the idea that is will be of use.

The map is said to ‘represent’ the sea, and taking any small enough part of the sea where water is represented by blankness, it would be as accurate as any map. Indeed the lines of longitude and latitude are merely arbitrary conventions. Again when looked at with disregard for any practical physical interpretation, but importantly with a strict and correct idea of reason, the idea of such a map is sound. It is only with the application to a real situation that it appears foolish, the nonsense orld of Carroll’s literature is here conceived from a logical base.

Lewis Carroll’s restless mind was always eager to invent new games and objects, he apparently invented the original double-sided adhesive tape and designed an early form of Scrabble (Fisher page 12). It is generally considered that Carroll modelled his White Knight character in Through the Looking-Glass upon himself, and they clearly share a passion for inventions and tricks. One example of this, which incorporates Carroll’s eagerness to include situations where gravity behaves unusually, is the White Knight’s charming invention of “… a new way of getting over a gate”.